I am posting my comment above as an answer. There are two different questions: does there exist a section that is *generically* transversal to the foliation, and does there exist a section that is *everywhere* transversal to the foliation.

The second question has a negative answer. Indeed, for a compact Riemann surface $C$ of positive genus, consider the product $\mathbb{CP}^1$-bundle, $C\times \mathbb{CP}^1$. For the second projection, $$\text{pr}_2:C\times \mathbb{CP}^1,$$ the fibers of $\text{pr}_2$ form the leaves of an everywhere regular foliation $\mathcal{F}$ on $C\times \mathbb{CP}^1$. Every cross-section of $\text{pr}_1$ is the graph of a morphism, $$f:C\to \mathbb{CP}^1.$$ This cross-section is transversal to $\mathcal{F}$ at precisely the image of the submersive locus of $f$. An everywhere submersive holomorphic map between compact, connected Riemann surfaces is an unbranched cover. Since $\mathbb{CP}^1$ is simply connected, there is no such unbranched cover. Thus, there is no cross-section that is everywhere transversal to $\mathcal{F}$.

On the other hand, there always exist sections that are generically transversal to $\mathcal{F}$. Indeed, one consequence of Tsen's theorem is that every $\mathbb{CP}^1$-bundle is bimeromorphic to a product family. Every (saturated) meromorphic foliation on the product family is a regular holomorphic foliation away from finitely many fibers. For a regular fiber $\{x\}\times \mathbb{CP}^1$, for every point $t\in \mathbb{CP}^1$, the fiber of the foliation at this point is a $1$-dimensional subspace $L$ of the $2$-dimensional vector space $T_{C,x}\oplus T_{\mathbb{CP}^1,t}$.

By Riemann-Roch, etc., there exists a nonconstant morphism from $C$ to $\mathbb{CP}^1$ that is submersive at $x$. Since the automorphism group of $\mathbb{CP}^1$ is transitive, we can post-compose to arrange that $x$ maps to $t$. Finally, since the automorphism group of $(\mathbb{CP}^1,x)$ acts transitively on $T_{\mathbb{CP}^1,x}\setminus\{0\}$, we can post-compose further and arrange that the graph of the morphism $$f:(C,x)\to (\mathbb{CP}^1,t),$$
at $(x,t)$ has tangent space that is different from $L$. Thus, the graph of $f$ gives a cross-section that is transversal at one point. The transversal locus is Zariski open, hence the transversal locus is the section.